is a probability on R.
(10%)
for n=1,2,... Why does the monotone convergence theorem fail here? (15%)
and F(x)
be the characteristic function and distribution function of a random variable
X, respectively.
For any two real numbers a and b, a<b,
which are continuity points of
F(x), show that
(20%)
,
in accordance with a Poisson process having rate 
.
Upon arrival, a customer is immediately served by one of an infinite number
of possible servers, and the service times are independent random variables
having a common distribution function
G. 
and
Yi~
.
In the case of 
,
find the large-sample distribution of the random variable , where 
,
as the sample size n
approaches infinity. (15%)
before breaking down; and suppose that it takes an exponential amount of
time having mean 
to repair the machine. If the machine is in working condition at time 0,
then what is the probability that it will be working at time t=10.
(20%)--- END ---